How do you find two positive numbers whose sum is 300 and whose product is a maximum?

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Answer 1 · 2015-02-22T11:50:39

The answer is that both of the numbers have to be $150$ $150$ .

Let $x$ $x$ is one of the two number, the other one is $300 - x$ $300-x$ , so the function product is:

$y = x (300 - x) \Rightarrow y = 300 x - x^{2}$ $y=x(300-x)rArry=300x-x^2$ , abd now let's find the maximum of the function:

$y'=300-2x$ that is positive in $(-oo,150)$ and zero in $150$ .

So, before 150 the function grows, and after $150$ the function decreases.

So the number $150$ is the local maximum of the function.